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Chow group of a product

Let $X$ and $Y$ be smooth varieties over $k$. I was wondering if there is a decomposition of the Chow group $CH(X\times Y)$ in terms of $CH(X)$ and $CH(Y)$ similar to the Kunneth decomposition of ...
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97 views

What can one say about zero-cycle groups for products of Chow motives

What can one say about the Chow group of zero-cycles (up to rational equivalence) for a product of smooth projective varieties and Chow motives (so, I am interested in the kernel $Chow_0(P)\otimes ...
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1answer
145 views

Deformation space form the point of view of intersection theory

I'm interested in deformations of subvarieties of a toric variety $X$. Suppose we know a subvariety $V$ in the Chow group of $X$, for example, $V$ is a linear combination of powers of hypersurfaces. ...
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2answers
258 views

on the Zariski sheafification of Quillen's K-theory

Let $X$ be a smooth k-variety and denote by $K_n$ Quillen's K-theory sheaf, that is: the Zariski sheaf on $X$ associated to the presheaf $U \mapsto K_n(U)$. The Bloch-Quillen formula says that ...
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1answer
834 views

For which varieties is the natural map from the Chow ring to integral cohomology an isomorphism?

My apologies if this question is too naive. Let $X$ be a smooth projective complex variety. There is a natural map $A^{\bullet}(X) \to H^{2\bullet}(X)$ of graded rings from the Chow ring of $X$ to ...
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1answer
120 views

Can generalization of Mumford’s theorem imply Mumford’s theorem for surface?

Mumford’s theorem for surface says that for a surface $S$ with $p_g(S)\neq0$ ,$\text{CH}_0(S)$ is not representable(or infinite-dimensional). But in Voisin's LECTURES ON THE HODGE ...
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1answer
213 views

About generalized Bloch conjecture

Conjecture((generalized Bloch conjecture)): Suppose that $\text{H}^{k,0}(X)=0$ for all $k>0$. Then $\text{CH}_0(X)=\mathbb{Z}.$ Is generalized Bloch conjecture known for complete intersections of ...
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295 views

Commutativity of the Chow ring in positive characteristic

I was looking in Ravi Vakil's notes on Intersection Theory, Class 20, where he introduces the bivariant intersection theory, in particular the Chow ring $A^\ast (X)$. On p. 2, he writes the following ...
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511 views

problematic proof of the moving lemma, second part ?

I have heard several times that the proof of the second part of the Chow's moving lemma (of algebraic geometry), is problematic; and that this is the reason Fulton, Intersection theory, does not use ...
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211 views

Vanishing of motivic cohomology with finite coefficients in negative degrees

I wonder whether the "finite coefficient version" of Beilinson-Soule conjecture i.e. the following statement holds or not. STATEMENT: Let $X$ be a smooth and projective scheme over a finite field ...
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1answer
289 views

Chow group of a (particular) motive [+ reference request]

I have two (not unrelated) questions. Let me first give a short introduction. Introduction For a general overview of the setup I refer to the introduction (§1) of [Zhang]. Let $k$ be a number field ...
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1answer
238 views

Is There a Mayer-Vietoris Spectral Sequence of Motivic Cohomology for Closed Coverings?

For etale cohomology, there is a spectral sequence of the following form ("Mayer-Vietories spectral sequence for closed covers"): $E_{1}^{p,q}=\oplus_{i_{0}< \cdots < i_{p}} H_{ Y_{i_{0} \cdots ...
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1answer
252 views

examples of Chow rings of surfaces

Can somone provide me (articles/literature) with examples of Chow rings of surfaces? (e.g. here: http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf Chapter 9) What I want is a list of (smooth ...
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1answer
159 views

Putting two complete varieties in a family over the projective line

Let $X$ and $Y$ be two proper varieties of dimension $n$ over a field $k$. I'm looking for "reasonable" conditions, under which, there exists a proper and dominant morphism $f:V\to \mathbb{P}^1_k$, ...
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1answer
209 views

Schubert problems to cycle class in Grassmanian

Say I have a family of linear spaces, and that I can solve all Schuber problems of that family (that is, how many members of the family pass through a set $S$ of linear spaces, where we consider all ...
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175 views

Generalized linear systems

Let $X$ be an algebraic variety and let $Z\subset X$ be a subvariety. Let $[Z]$ be the class defined by $Z$ in the Chow group. Let $L(Z)$ be set of effective algebraic cycles on $X$ linearly ...
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1answer
340 views

Zero-cycles on an arithmetic surface

Could anyone give a reference for the following statement, which I believe is true. "Let X be a regular scheme, flat over $Spec( \mathbb{Z}) $, with fiber dimension $1$. Then the Chow group $CH^2(X)$ ...
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1answer
496 views

Are Chow groups generated by local complete intersections?

Let $X$ be a smooth projective variety over an algebraically closed field. The Chow group $\mathbb Q\mathrm{CH}^d(X)$ is $\mathbb Q$--linearly generated by irreducible subvarieties $Z \subseteq X$ of ...
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1answer
395 views

Do we know the Chow groups of spheres?

Let $k$ be a field (of char. not $2$) and $X_k=\text{Spec} (k[x_1,\cdots,x_n]/(x_1^2+\cdots +x_n^2-1))$. Do we know the Chow groups $A_i (X_k)$? I could not find any references, even for $X_{\mathbb ...